A Proof of the Collatz Conjecture via Translation and Structuralization of Natural Numbers and the Collatz Map
Authors/Creators
- 1. Independent Researcher
Description
In this paper, we treat the Collatz conjecture not as a problem of tracing natural numbers one by one, but as a structural problem obtained by translating positive odd integers into alternating binary notation (ABN) and Collatz phase expressions (CPE). In this framework, the accelerated Collatz map $$F(n)=\frac{3n+1}{2^{\nu_2(3n+1)}} \quad (n \text{ odd})$$ is described as a local rewriting rule on structural expressions.
First, we construct reversible correspondences among positive odd integers, alternating sums of Mersenne numbers, canonical ABN, and canonical CPE. We then derive one-step inequalities for the structural quantities μ, H_CS, H_K, and B from a local analysis of the 27 endpoint patterns. These inequalities provide a way to control the structural complexity of a Collatz step without following each integer orbit directly.
The main argument separates the proof into a high-μ region and a low-μ region. In the high-μ region, the second case is forced into a completely frozen state. Using the identity $$\nu_2(3n+1)=k_1(n)$$ together with a digit-length estimate, the frozen case is reduced to a case split on k_1. If k_1 >= 3, the digit length strictly decreases; if k_1 = 2, the digit length is nonincreasing while the numerical value strictly decreases. The remaining boundary case k_1 = 1 is handled together with terminal normalization corrections.
In the low-μ region, the case μ = 1 is resolved directly. For μ = 2 and μ = 3, the one-step images are reduced to finitely many ABN/CPE types of total digit length at most 27 after a controlled truncation of zero blocks. The remaining verification is not carried out on ordinary integer orbits, but on a finite-type map obtained by applying the accelerated Collatz map and then truncating each zero block to the three representative types K(1), K(2), and K(>=3). This gives a finite check over a finite set. Branches that temporarily enter the high-μ region are returned to the high-μ descent theorem and hence also resolve in finite time.
Combining the high-μ descent with the finite-type verification in the low-μ region, every positive odd state is resolved. Since every positive integer reaches a positive odd state after dividing by powers of 2, it follows that every Collatz orbit of a positive integer reaches 1 after finitely many steps.
Notes (English)
This version updates the English manuscript to correspond to the latest Japanese version v190.
The main revision is the strengthening of the high-mu descent argument. The proof now uses complete freezing, the identity nu_2(3n+1)=k_1(n), and a case split on k_1. In the frozen high-mu case, k_1 >= 3 gives a strict decrease of digit length, k_1 = 2 gives nonincrease of digit length together with strict decrease of the integer value, and the boundary case k_1 = 1 is handled by terminal normalization corrections.
The low-mu part has also been updated. The cases mu = 1, mu = 2, and mu = 3 are reduced to a finite-type verification after controlled truncation of zero blocks. The finite check is performed not on ordinary integer orbits, but on a finite-type map in which each zero block is truncated to the representative types K(1), K(2), and K(>=3).
This version also restores and updates several propositions, examples, lemmas, labels, and references so that the English manuscript matches the Japanese v190 structure. The 27 local endpoint-pattern proposition, the tail-normalization example, and the two-type tail-normalization lemma are included. The manuscript has been checked for untranslated Japanese text, undefined references, duplicate labels, and placeholder expressions, and it compiles successfully with XeLaTeX.
Notes (English)
We revised the proof in two main places. First, we corrected and sharpened the one-step estimate for $H_K$, explaining the bound
$$
\Delta H_K \le 2 - 2\mu_C(E) + H_C(E)
$$
directly from the local $27$-pattern analysis. Second, on the low-$\mu$ side, we reorganized the argument around a notion of low-$\mu$ resolution, proved its invariance under radius-$3$ truncation using the previously established carry-reach bound, and rewrote the finite-check part as
$$
\text{truncation invariance} ;\to; \text{representative covering} ;\to; \text{finite search} ;\to; \text{general low-}\mu \text{ resolution}.
$$
We also updated the abstract, introduction, proof overview, and conclusion so that they match this revised structure.
Notes
A separate English final checklist for version v25 has also been prepared. It summarizes the items checked before release, including correspondence with the Japanese v190 manuscript, translation consistency, mathematical notation, labels and references, compilation status, remaining warnings, and Zenodo submission metadata. The checklist is intended as an auxiliary release document and does not constitute an additional mathematical argument.
Files
Main_Collatz_merged_EN_v25.pdf
Additional details
Related works
- Is identical to
- Preprint: 10.17605/OSF.IO/TXZYA (DOI)
References
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